Class 12 Notes - Continuity and Differentiability

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Continuity and Differentiability

This chapter explores the fundamental concepts of continuity and differentiability of functions, which are essential for understanding calculus. Students will learn how to determine if a function is continuous or differentiable at a point and over an interval, and how these properties are related.

1. Continuity

  • A function f(x) is continuous at a point x = a if:
    • f(a) is defined
    • limx→a f(x) exists
    • limx→a f(x) = f(a)
  • A function is continuous in an interval if it is continuous at every point in that interval.
  • Types of discontinuities: removable, jump, and infinite.

Example:

Is the function f(x) = x2 continuous at x = 2?

Solution:
f(2) = 4
limx→2 x2 = 4
Since both are equal, f(x) is continuous at x = 2.

2. Differentiability

  • A function f(x) is differentiable at x = a if the derivative f'(a) exists.
  • If a function is differentiable at a point, it is also continuous at that point (but not vice versa).
  • Left-hand derivative and right-hand derivative must be equal for differentiability.

Example:

Is f(x) = |x| differentiable at x = 0?

Left-hand derivative: -1
Right-hand derivative: +1
Since they are not equal, f(x) = |x| is not differentiable at x = 0.

3. Important Results

  • Every differentiable function is continuous, but every continuous function may not be differentiable.
  • Algebra of continuous and differentiable functions (sum, difference, product, quotient).
  • Chain rule for differentiation.

4. Practice Problems

  1. Check the continuity of f(x) = 1/x at x = 0.
  2. Is f(x) = x3 differentiable at x = 0?
  3. Find the points of discontinuity of f(x) = [x] (greatest integer function).
  4. Show that f(x) = sin(x) is continuous and differentiable everywhere.
  5. If f(x) = x2 + 3x + 2, find f'(x).

5. Summary

  • Continuity: No breaks, jumps, or holes in the graph of the function.
  • Differentiability: The function has a defined tangent (slope) at the point.
  • Differentiability implies continuity, but not the other way around.